## Linear Algebra and Its Applications, exercise 1.3.5

Exercise 1.3.5. Given the following system of equations: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcr}x&-&y&=&0 \\ 3x&+&6y&=&18 \end{array}$

find a solution to the system. (The exercise also calls for doing a sketch of the lines corresponding to the equations, as well as the line corresponding to the second equation after elimination. I’m skipping that part.)

Answer: We subtract 3 times the first equation from the second equation: $\setlength\arraycolsep{0.2em}\begin{array}{rcrcr}x&-&y&=&0 \\ 3x&+&6y&=&18 \end{array} \Rightarrow \begin{array}{rcrcr}x&-&y&=&0 \\ &&9y&=&18 \end{array}$

We can then back-substitute to solve for $y$ and then $x$: $\begin{array}{rcrcr}9y = 18&\Rightarrow&y = 2 \\x - y = 0&\Rightarrow&x = 2 \end{array}$

So the solution is $x = 2$, $y = 2$.

The first equation corresponds to the line $y = x$ at a 45 degree angle to the $x$ axis and intersecting the origin. The second equation corresponds to the line $y = -\frac{1}{2}x + 3$ with slope $-\frac{1}{2}$ and intersecting the $y$ axis at the point (0, 3). After elimination the second equation $9y = 18$ (or $y = 2$) corresponds to a horizontal line intersecting the $y$ axis at the point (0, 2).

UPDATE: Corrected the $y$-intercept for the second equation; thanks go to pana8a7 for finding and fixing this error. Also corrected a type in the elimination steps (had 19 instead of 18). Finally, took the time to convert all variable names and in-line equations to use LaTeX.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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### 2 Responses to Linear Algebra and Its Applications, exercise 1.3.5

1. pana8a7 says:

3x+6y=18
3(x+2y)=18
x+2y=6
y=-x/2+3 not y=-x/2+6

• hecker says:

You are correct. Thank you for finding and fixing this error! I have updated the post (and found and fixed another error at the same time).